Calabi-Yau Pairs and Mirror Symmetry for Fano Varieties

几十年来，代数几何一直是数学的核心学科之一，作为新思想和重要问题的来源，这门学科在今天仍然和150年前一样重要。代数几何研究的基本对象是环境射影空间中由多项式方程定义的几何形状。最小模型程序表明，在手术前，这些形状可以由纯几何类型的“积木”构成，这些形状有：(1)正曲率(Fano变种)，(2)零曲率(Calabi-Yau变型)，(3)负曲率(一般类型的变种)。这些纯几何类型直观地对应于球面、欧几里德平面和双曲平面的几何。每一纯类型的几何形状都有不同的特征和属性：有理曲线的存在，家族行为，品种与其他品种之间的手术操作。在最近的发展中，一种非常有用的技术是考虑这些纯几何类型的扰动。扰动几何类型是为位于其上的各种形状和一些较低维度的形状(例如，原始形状的切片)的对定义的。扰动纯几何类型的变种称为对数Fano、对数Calabi-Yau和对数一般类型。它们共享关联纯类型的许多功能。配对的几何类型非常丰富：配对可以具有某种扰动类型(例如，log Calabi-Yau)，而其潜在的变种具有不同的纯几何类型(在我们的例子中为Fano)。然后，这对几何融合了Calabi-Yau和Fano几何的特征。我的研究集中在其几何或扰动几何是Fano或Calabi-Yau类型的变种和对。这些在数学物理中很重要：根据弦理论，物理中的基本物体是弦，而不是点状粒子。这些弦在背景中运动，除了空间和时间之外，还有额外的隐藏维度卷曲在不同的背景中，这是Fano或Calabi-Yau(取决于理论的版本)。显然，这个建议涉及到对数Calabi-Yau和Fano形状的几何。第一个项目研究保持附加不变量的对数Calabi-Yau形状的变换。最有趣的例子是由Fano形状和Calabi-Yau切片组成的对数Calabi-Yau形状之间的变换。然后需要进行变换以保持Calabi-Yau切片的体积。第二个项目明确地关注Fano簇的镜像对称性。镜像对称性是一种二元性，当两个数学上不同的背景几何图形产生相同的物理时，这两个几何图形就被称为镜像对偶。推测，Fano形状对所谓的簇种是镜像对称的。簇变种是原木Calabi-Yau形状的家族，可以通过第一个项目中研究的转换来粘合。拟议的研究将应用代数几何和最小模型程序中的技术和思想来加深我们对簇种和对数Calabi-Yau几何的理解。反过来，这项工作的结果将为Fano形状的几何提供一个新的角度。

Algebraic geometry has for many decades been one of the core disciplines of mathematics, and the subject remains as vital today as it was 150 years ago as a source of new ideas and important problems. The basic objects studied in algebraic geometry are geometric shapes defined by polynomial equations in an ambient projective space. The Minimal Model Program (MMP) shows that, up to surgery, these shapes can be constructed out of "building blocks" of pure geometric type, and these have:(1) positive curvature (Fano varieties),(2) zero curvature (Calabi-Yau varieties),(3) negative curvature (varieties of general type).These pure geometric types correspond intuitively to the geometry of the sphere, of the Euclidian plane and of the hyperbolic plane. The geometry of each pure type has distinct features and properties: presence of rational curves, behaviour in families, surgery operations between the variety and other varieties.. A very useful technique in recent developments has been to consider perturbations of these pure geometric types. The perturbed geometric types are defined for pairs of a variety and some lower dimensional shape lying on it (for example a slice of the original shape). The varieties of perturbed pure geometric types are called log Fano, log Calabi-Yau, and of log general type. They share many of the features of the associated pure types. The geometry of pairs is very rich: a pair can have a certain perturbed type (log Calabi-Yau for example) while its underlying variety has a different pure geometric type (Fano in our example). The geometry of the pair then blends features of Calabi-Yau and Fano geometries. My research concentrates on varieties and pairs whose geometry or perturbed geometry is of Fano or Calabi-Yau type. These are important in mathematical physics: according to string theory, the fundamental objects in physics are strings rather than point-like particles. These strings move in a background that, in addition to space and time, has extra hidden dimensions curled up in a background variety which is Fano or Calabi-Yau (depending on the version of the theory). Explicitly, this proposal is concerned with the geometry of log Calabi-Yau and Fano shapes. The first project studies transformations of log Calabi-Yau shapes that preserve an additional invariant. The most interesting case is that of transformations between log Calabi-Yau shapes that are made of a Fano shape and a Calabi-Yau slice of it. The transformations are then required to preserve the volume of the Calabi-Yau slice. The second project focusses explicitly on mirror symmetry for Fano varieties. Mirror symmetry is a duality that occurs when two mathematically different background geometries produce the same physics; the two geometries are then called mirror dual. Conjecturally, Fano shapes are mirror symmetric to so called cluster varieties. Cluster varieties are families of log Calabi-Yau shapes that can be glued by the transformations studied in the first project. The proposed research will apply techniques and ideas from algebraic geometry and from the Minimal Model Program to deepen our understanding of cluster varieties and of log Calabi-Yau geometries. In turn, the results of this work will provide a new angle on the geometry of Fano shapes.